A Law of Iterated Logarithm on Lamplighter Diagonal Products

We prove a Law of Iterated Logarithm for random walks on a family of diagonal products constructed by Brieussel and Zheng (2021). This provides a wide variety of new examples of Law of Iterated Logarithm behaviours for random walks on groups. In particular, it follows that for any $\frac{1}{2}\leq \beta\leq 1$ there is a group $G$ and random walk $W_n$ on $G$ with $\mathbb{E}|W_n|\simeq n^\beta$ such that $$0<\limsup \frac{|W_n|}{n^\beta(\log\log n)^{1-\beta}}<\infty$$ and $$0<\liminf \frac{|W_n|(\log\log n)^{1-\beta}}{n^\beta}<\infty.$$